Norm-free event triggered information exchange for distributed control of multiagent systems

ABSTRACT

Methods and systems for information exchange of a vehicle in a multiagent system are disclosed. The methods and systems include: receiving one or more neighboring states broadcast by one or more neighboring vehicles; transmitting a last broadcast state of the vehicle to the one or more neighboring vehicles; determining a current state of the vehicle based on the one or more neighboring states and the last broadcast state; determining a norm-free information exchange triggering condition based on the last broadcast state, the current state, and an estimated command; and in response to the current state violating the norm-free information exchange triggering condition, transmitting the current state to the one or more neighboring vehicles. Other aspects, embodiments, and features are also claimed and described.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims the benefit of U.S. Provisional PatentApplication Ser. No. 63/261,325, filed Sep. 17, 2021, the disclosure ofwhich is hereby incorporated by reference in its entirety, including allfigures, tables, and drawings

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This invention was made with government support under the UniversalTechnology Corporation Grant 162642-20-25-C1 awarded by the Air ForceResearch Laboratory Aerospace Systems Directorate. The government hascertain rights in the invention.

SUMMARY

The following presents a simplified summary of one or more aspects ofthe present disclosure, to provide a basic understanding of suchaspects. This summary is not an extensive overview of all contemplatedfeatures of the disclosure and is intended neither to identify key orcritical elements of all aspects of the disclosure nor to delineate thescope of any or all aspects of the disclosure. Its sole purpose is topresent some concepts of one or more aspects of the disclosure in asimplified form as a prelude to the more detailed description that ispresented later.

In some aspects of the present disclosure, methods, systems, andapparatus for information exchange of a vehicle in in a multiagentsystem are disclosed. These methods, systems, and apparatus can includesteps or components for: receiving one or more neighboring statesbroadcast according to an undirected and connected graph topology by oneor more neighboring vehicles; transmitting a last broadcast state of thevehicle to the one or more neighboring vehicles; determining a currentstate of the vehicle based on the one or more neighboring states and thelast broadcast state; determining a norm-free information exchangetriggering condition based on the last broadcast state, the currentstate, and an estimated command; and in response to the current stateviolating the norm-free information exchange triggering condition,transmitting the current state to the one or more neighboring vehicles.

These and other aspects of the disclosure will become more fullyunderstood upon a review of the drawings and the detailed description,which follows. Other aspects, features, and embodiments of the presentdisclosure will become apparent to those skilled in the art, uponreviewing the following description of specific, example embodiments ofthe present disclosure in conjunction with the accompanying figures.While features of the present disclosure may be discussed relative tocertain embodiments and figures below, all embodiments of the presentdisclosure can include one or more of the advantageous featuresdiscussed herein. In other words, while one or more embodiments may bediscussed as having certain advantageous features, one or more of suchfeatures may also be used in accordance with the various embodiments ofthe disclosure discussed herein. Similarly, while example embodimentsmay be discussed below as devices, systems, or methods embodiments itshould be understood that such example embodiments can be implemented invarious devices, systems, and methods.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an example multiagent system according to some embodiments.

FIG. 2 is a numerical example of the disclosed approach or method whileexchanging sampled rate data points with a first learning rate accordingto some embodiments.

FIG. 3 is a numerical example of the disclosed approach or method whileexchanging local solution-predictor curves with the first learning rateaccording to some embodiments.

FIG. 4 is a numerical example of the disclosed approach or method whileexchanging sampled rate data points with a second learning rateaccording to some embodiments.

FIG. 5 is a numerical example of the disclosed approach or method whileexchanging local solution-predictor curves with the second learning rateaccording to some embodiments.

FIG. 6 is a block diagram conceptually illustrating a multiagent systemfor information exchange according to some embodiments.

FIG. 7 is a flowchart illustrating an example process for informationexchange of a vehicle in a multiagent system according to someembodiments.

DETAILED DESCRIPTION

The detailed description set forth below in connection with the appendeddrawings is intended as a description of various configurations and isnot intended to represent the only configurations in which the subjectmatter described herein may be practiced. The detailed descriptionincludes specific details to provide a thorough understanding of variousembodiments of the present disclosure. However, it will be apparent tothose skilled in the art that the various features, concepts andembodiments described herein may be implemented and practiced withoutthese specific details. In some instances, well-known structures andcomponents are shown in block diagram form to avoid obscuring suchconcepts.

Multiagent systems are computerized systems that comprise multipleinteracting intelligent agents. The interacting agents may be used, forexample, to solve problems that are difficult or impossible for anindividual agent or a monolithic system to solve. Examples of multiagentsystems are groups of interacting aerial drones, and other roboticagents that interact with one another.

In some multiagent systems, control of the interacting agents is donevia a distributed control scheme. Reducing the agent-to agentinformation exchange in a distributed control of a multiagent system canbe achieved by a norm-free and adaptive event-triggering rule for eachagent, where the multiagent system is decentralized and predicated on asolution-predictor curve method. The decentralized feature provides theproposed event-triggering rule to depend on its own error signals of anagent without acquiring any neighboring or global information. Thenorm-free feature provides a left side of the proposed event-triggeringrule inequality to not be dependent on absolute values of error signalsalso known as distance, to enable reduction of agent-to-agentinformation exchange.

Traditionally, a local nature of information exchange between agents isappreciated in a feedback control more than a global nature ofinformation exchange. The significantly increased communication, reducedsecurity, and low feasibility between all agents makes global nature ofinformation exchange less desirable in the development of distributedcontrol architectures. Distributed control architectures comprise acontinuous agent-to-agent information exchange or a periodicagent-to-agent information exchange which may result in network overloadand excessive usage of energy sources of each agent. An event-triggeredcontrol theory mitigates excessive use of energy sources and networkoverload of the continuous and/or periodic information exchange in anaperiodic and asynchronous way. The control architectures of the eventtriggered control theory includes a norm (e.g., absolute values orEuclidian norms of error signals). Reducing the norm or making thesystem norm-free enables a smaller number of events and reduction ininformation change.

In the present disclosure, methods and systems including a norm-free andan adaptive event triggering rule can reduce agent-to-agent informationexchange in distributed control of multiagent systems. Specifically, thenorm free and the adaptive event triggering rule can be decentralizedand predicated on the solution-predictor curve method. In order tointroduce norm free and decentralized features, an adaptive term isutilized in the event triggering rule to estimate unknown variableunavailable to the agent. The adaptive term can be utilized in theevent-triggering rule for each agent to estimate an unknown variable.The unknown variable is a time-varying command and is available to asubset of agents (e.g., leader agents) to control all agents of themultiagent system accordingly to the time-varying command. Theestimation of the unknown variable is enabled by a solution-predictorcurve method. The solution-predictor curve method provides furtherreduction of agent-to-agent information exchange by allowing each agentto store the curve and exchanges the parameters when an event occurs ina distributed manner for approximating the solution trajectory of eachagent. The solution-predictor curve method enables reduction ofagent-to-agent information exchange where each agent stores the curveand exchange the parameters for approximating the solution trajectory ofeach agent. when an event occurs rather than standard sampled dataexchange.

A generic event triggering rule comprising absolute values is shownbelow in equation 1.

|a(t)|≤|b(t)|+c(t)  Equation 1

where |a(t)| and |b(t)| are absolute values with scalar properties, andc(t) being a non-negative-definite scalar. When the relationship betweena left side of the equation and a right side of the equation isviolated, a dynamic system sends new information. The absolute values of|a(t)| and |b(t)| are considered to be a norm, and the left side of theequation comprising the term |a(t)| may be replaced by a(t) to form anorm-free dynamic system. Then, the possible negative value of a(t) canresult in less number of events, and therefore, better informationexchange reduction since Equation 1 does not get violated forsufficiently large negative numbers of a(t). The norm-free featureenables the left side of the proposed event-triggering rule inequalityto be independent on distances such as absolute values of error signalsto allow better agent-to-agent information exchange reduction.

In the present disclosure,

,

₊, and

₊ respectively denote the sets of integers, positive integers, andnonnegative integers;

,

^(n), and

^(n×m), respectively denote the sets of real numbers, n×1 real vectors,and n×m real matrices;

,

₊,

₊ ^(n×m), and

₊ ^(n×m), respectively denote the sets of positive real numbers,nonnegative real numbers, positive-definite real matrices, andnonnegative-definite real matrices; 0_(n) and 1_(n) respectively denotethe n×1 vectors of all zeros and ones; and “

” denotes the equality by definition. Furthermore, (⋅)^(T) denotestranspose, (⋅)⁻¹ denotes inverse, ∥⋅∥₂ denotes the Euclidean norm, λ(A)and λ(A) respectively denote the minimum and maximum eigen values of thereal matrix A∈

^(n×m), diag(a) denotes the diagonal matrix with the real vector a∈

^(n) on its diagonal, and [A]_(ij) denotes the entry of the real matrixA∈

^(n×m) on the ith row and jth column, i=1, . . . , n, j=1, . . . , m.

A graph theory is a study of graphs that includes mathematicalstructures to model pairwise relations between objects. The graphs aremade of vertices or nodes that are connected by edges or lines and thegraph G can include a pair (V, E) where V is the set of vertices and Eis the set of edges. The graphs can include an undirected graph whereedges link two vertices symmetrically and a direct graph where edgeslink two vertices asymmetrically. In the present disclosure, undirectedgraph G is defined by V_(G)={1, . . . , n)}, which are set of nodes orvertices and E_(G)⊂V_(G)×V_(G), which are set of edges. The nodes i andj are neighbors when (i,j)∈E_(G) and the neighboring relationship isrepresented by i˜j. Additionally, the graph G is connected when there isa finite amount of path between i₀ . . . i_(L) with i_(k−1)˜i_(k) wherek=1, . . . , L between two distinct nodes. The graph includes a degreematrix provided by the given form as shown in equation 2.

$\begin{matrix}{{{D(G)}\overset{\bigtriangleup}{=}{{diag}(d)}},{d = \left\lbrack {d_{1},\ldots,d_{n}} \right\rbrack^{T}}} & {{Equation}2}\end{matrix}$

where d_(i) denotes the number of neighboring node i, diag(d) denotesthe diagonal matrix with a real vector on its diagonal,

denotes the equality by definition, and T denotes the transpose. Anadjacency matrix A(G)∈

^(n×n) of G has entries [A(G)]_(ij)=1 when (i,j)∈E_(G) and [A(G)]_(ij)=0otherwise. A Laplacian matrix of graph G is provided in equation 3 withrespect to multi agent systems over the connected and undirected graphG. In some examples, the present disclosure uses multiagent systems overthe connected and undirected graph G.

$\begin{matrix}{{\mathcal{L}(G)}\overset{\bigtriangleup}{=}{{D(G)} - {A(G)}}} & {{Equation}3}\end{matrix}$

A lemma is a proposition which is used as a steppingstone to a largerresult in informal logic mapping and may be referred to as a helpingtheorem or an auxiliary theorem. In the present disclosure, three lemmasare used. A first lemma or the Laplacian matrix includes a diagonalmatrix K=diag(k) with real vectors on its diagonal, k=[k₁, . . . ,k_(n)]^(T), k_(i)∈

₊, a set of nonnegative integers, i=1, . . . , n, and that at least onek_(i)≠0. The first lemma can be expressed by the given form as shown inequation 4 and 5.

$\begin{matrix}{{F(G)}\overset{\bigtriangleup}{=}{{{L(G)} + K} \in {\mathbb{R}}_{+}^{n \times n}}} & {{Equation}4}\end{matrix}$ $\begin{matrix}{{F^{- 1}(G)} \in {\mathbb{R}}_{+}^{n \times n}} & {{Equation}5}\end{matrix}$

where

₊ ^(n×n) denotes positive definite real matrices, and F⁻¹ denotesinverse of F.

A second lemma or the Young's inequality can be expressed by the givenform as shown in equation 6.

$\begin{matrix}{{x^{T}y} \leq {{\frac{\rho}{2}x^{T}x} + {\frac{1}{2\rho}y^{T}y}}} & {{Equation}6}\end{matrix}$

where x∈

^(n), y∈

^(m), and p∈

⁺.

^(n) denotes n×1 real vectors,

^(m) denotes m×1 real vectors, and

⁺ denotes set of real positive numbers.

A third lemma or the comparison principle includes a dynamical systemgiven by the equation 7 shown below.

{dot over ({circumflex over (v)})}(t)=f(t,{circumflex over (v)}(t)),{circumflex over (v)}(0)={circumflex over (v)} ₀  Equation 7

where {circumflex over (v)}(t)∈

and f(t,{circumflex over (v)}(t)) being continuous in time t and locallyLipschitz in {circumflex over (v)}(t) for all t∈

₊ and all {circumflex over (v)}(t)∈R⊂

. A maximum interval of existence of the solution {circumflex over(v)}(t) is [0, T) and {circumflex over (v)}(t)∈

for all t∈[0, T). Additionally, a continuous function is denoted as v(t)and satisfies the condition shown by the equation 8 shown below.

{dot over (v)}(t)≤f(t,v(t)), v(0)≤{circumflex over (v)} ₀  Equation 8

By applying the maximum interval of existence relationship including{circumflex over (v)}(t)∈R for all t∈[0, T) to equation 8, thecontinuous function is expressed by the given form as shown in equation9, and for all t∈[0, T), T may be infinity.

v(t)≤{circumflex over (v)}(t)  Equation 9

A connected and undirected graph G can include multiple agents that aredenoted by n. A leader agent can be a subset of the multiple agents andhas the knowledge of a bounded time-varying command c(t) that has apiecewise continuous function and bounded time rate of change. Afollower agent can be a remain set of the multiple agents that do nothave the knowledge of this command.

The dynamics of agent i, for i=1, . . . , n, is given by the equation 10shown below.

{dot over (x)} _(i)(t)=u _(i)(t), x _(i)(0)=x _(i0)  Equation 10

where x_(i)(t)∈

is the state and u_(i)(t)∈

is the control signal. The control signal of agent i, for i=1, . . . ,n, can be expressed by equation 11 shown below.

$\begin{matrix}{{u_{i}(t)} = {{- {\sum\limits_{i\sim j}\left( {{{\hat{x}}_{i}(t)} - {{\hat{x}}_{j}(t)}} \right)}} - {k_{i}\left( {{{\hat{x}}_{i}(t)} - {c(t)}} \right)}}} & {{Equation}11}\end{matrix}$

where k_(i)=1 for the leader agents and k_(i)=0 for the follower agents,and {circumflex over (x)}_(i)(t)∈

being the latest broadcast of x_(i)(t) of agent i for anevent-triggering approach. The following control signal would enable allagents to approach the command with reduced agent-to-agent informationexchange.

The unknown variable providing the time-varying command to the leaderagent allows an adaptive event-triggering rule to reduce agent-to agentinformation exchange by providing an estimate of the command. Theadaptive event triggering rule for agent i, for i=1, . . . , n, isexpressed by the given form as shown in equation 12.

(x _(i)(t)−ŵ _(i)(t))(x _(i)(t)−{circumflex over (x)} _(i)(t))≤μ((x_(i)(t)−ŵ _(i)(t))²  Equation 12

where μ∈[0,1) and μ denotes an event-triggering threshold constant andŵ_(i)(t) is expressed by equation 12 as(shown below.

{dot over (ŵ)}_(i)(t)=−γ_(i)((x _(i) t)−{circumflex over (x)}_(i)(t))+2μ(ŵ _(i)(t)−x _(i)(t))), ŵ _(i)(0)=ŵ _(i0)   Equation 13

where γ_(i)∈

₊ and γ_(i) denotes the adaptation gain. ŵ_(i)(t) is the adaptive termutilized to provide an estimate of command c(t) for ŵ_(i)(t)∈

.

Referring back to equation 1, the equation resembles similarconfiguration of equation 12 including the left side of the equation andthe right side of the equation. The left side of equation 12 does notdepend on the absolute values of the error signals. For example, if theleft side of the equation 12 is negative, it is not possible to violatethe relationship, which enables better information exchange reduction.The only condition to violate the relationship takes place when the leftside of the equation 12 is positive and is larger than the ride side ofthe equation 12. Therefore, the event-trigger rule provided by equation12 may be broadened and considered for the proposed scenarios.

A first scenario is the standard sampled data exchange method thatdepends on the zero-order-hold operator. A first method of the firstscenario describes for agent i, for i=1, . . . , n, broadcasts a sampleddata of a state value denoted as x_(i)(t) through a zero-order-holdoperator to its neighbors when the event-triggering rule given byequation 12 is violated when {circumflex over (x)}_(i)(t)=x_(i)(t_(di))for t∈t_(di), t_((d+1)i). To state a second scenario predicated on thesolution-predictor curve method,

${{\hat{x}}_{i}(t)}\overset{\bigtriangleup}{=}{x_{i}(t)}$

(since each agent has continuous access to its own state),

${{\hat{x}}_{j}\overset{\bigtriangleup}{=}{x_{j}\left( t_{j}^{e} \right)}},{{{and}r}\overset{\bigtriangleup}{=}{c\left( t_{i}^{e} \right)}}$

can be used where t_(i) ^(e) is the time when equation 12 is violatedfor agent i and t_(j) ^(e) is the time when equation 12 is violated foragent j subject to i˜j. The trajectory agent i can be approximated byexplicitly solving equation 10 and equation 11 in time domain. Solvingequation 10 and 11, a local solution-predictor curve agent i, for i=1, .. . , n, satisfies

$\begin{matrix}{{\xi_{i}(t)}\overset{\bigtriangleup}{=}{{{\exp\left( {Q_{i}\left( {t - t_{i}^{e}} \right)} \right)}{x_{i}\left( t_{i}^{e} \right)}} + {\frac{B_{i}}{Q_{i}}\left( {1 - {\exp\left( {Q_{i}\left( {t - t_{i}^{e}} \right)} \right)}} \right)}}} & {{Equation}14}\end{matrix}$ where $\begin{matrix}{{B_{i}\overset{\bigtriangleup}{=}{{\sum_{i\sim j}{\hat{x}}_{j}} + {k_{i}r}}},{Q_{i} = {k_{i} + d_{i}}}} & {{Equation}15}\end{matrix}$

Referring to equation 14, each agent i stores for all of its neighborsand before the next even occurs, agent i can use for i˜j instead forusing the last received sampled data.

A second scenario can be predicated on the solution-predictor curvemethod. In the second scenario, agent i, for i=1, . . . , n, canbroadcast a solution-predictor curve of a state value denoted asx_(i)(t) to its neighbors when the event-triggering rule given byequation 12 is violated when {circumflex over (x)}_(i)(t)=ξ_(i)(t) fort∈t_(di), t_((d+1)i). In contrast to standard sampled data exchangemethod from the first scenario, the solution-predictor curve method hasthe ability to further reduce agent-to-agent information exchange sincethe solution predictor curve method does not rely on a constant butrather relies on a time varying {circumflex over (x)}_(i)(t), whichapproximates the solution of agent i, for i=1, . . . , n.

FIG. 1 illustrates an example of a multiagent system 100 consisting of afirst agent 104, a second agent 108, a third agent 112, and a fourthagent 116 which graphically illustrates the event-triggering scenariosfor the first and second method. An agent-to-agent information exchange120 can be predicated on event-triggering scenarios stated in the firstscenario. The first agent 104 is the leader agent, and the other agents108, 112, 116 are the follower agents.

A distributed control architecture can be provided by equation 11 of thepresent disclosure along with the proposed norm-free and adaptive eventtriggered rule provided by equation 12 and 13. Based on equation 11, 12and 13, a system-theoretical analysis for the sampled data exchange caseof the first method and the data exchange predicated on thesolution-predictor curve of the second method is described below byincorporating equation 10 into equation 11:

$\begin{matrix}{{{\overset{.}{x}}_{i}(t)} = {- {\sum\limits_{i\sim j}\left( {{x_{i}(t)} - {x_{j}(t)} - {k_{i}\left( {{x_{i}(t)} - {c(t)}} \right)} + {\sum\limits_{i\sim j}\left( {\left( {{x_{i}(t)} - {{\hat{x}}_{i}(t)}} \right) - \left( {{x_{j}(t)} - {{\hat{x}}_{j}(t)}} \right)} \right)} + {k_{i}\left( {{x_{i}(t)} - {{\hat{x}}_{i}(t)}} \right)}} \right.}}} & {{Equation}16}\end{matrix}$

The error between the state of agent i, for i=1, . . . , n and thecommand is described a

${e_{i}(t)}\overset{\bigtriangleup}{=}{{x_{i}(t)} - {{c(t)}.}}$

Incorporating the error between the state agent and the command toequation 16, a time derivative of error is described as:

$\begin{matrix}{{{\overset{.}{e}}_{i}(t)} = {{\sum\limits_{i\sim j}\left( {{e_{i}(t)} - {e_{j}(t)}} \right)} - {k_{i}{e_{i}(t)}} + {\sum\limits_{i\sim j}\left( {{\left( {{x_{i}(t)} - {{\hat{x}}_{i}(t)} - \left( {{x_{j}(t)} - {{\hat{x}}_{j}(t)}} \right)} \right) + {k_{i}\left( {{x_{i}(t)} - {{\hat{x}}_{i}(t)}} \right)} - {\overset{.}{c}(t)}},{{e_{i}(0)} = e_{i0}}} \right.}}} & {{Equation}17}\end{matrix}$

where e_(i0)=x_(i0)−c(0). In view of equation 4, with K=diag(k) andk=[k₁ . . . k_(n)]^(T), equation 17 can be further described as:

ė _(i)(t)=−F(G)e(t)+G(G)(x(t)−{circumflex over (x)}(t))−ρ(t), e(0)=e₀  Equation 18

where

${{x(t)}\overset{\bigtriangleup}{=}{\left\lbrack {{x_{1}(t)},\ldots,{x_{n}(t)}} \right\rbrack^{T} \in {\mathbb{R}}^{n}}},{{\hat{x}(t)}\overset{\bigtriangleup}{=}{\left\lbrack {{x_{1}(t)},\ldots,{x_{n}(t)}} \right\rbrack^{T} \in {\mathbb{R}}^{n}}},{{\rho(t)}\overset{\bigtriangleup}{=}{1_{n}{\overset{.}{c}(t)}}},{{{and}e_{0}}\overset{\bigtriangleup}{=}{\left\lbrack {{e_{10}(t)},\ldots,{e_{1n}(t)}} \right\rbrack^{T} \in {{\mathbb{R}}^{n}.}}}$

The error between the adaptive term and command is defined as

${{\overset{˜}{w}}_{i}(t)}\overset{\bigtriangleup}{=}{{{\overset{\hat{}}{w}}_{i}(t)} - {c(t)}}$

and is described as:

{dot over ({tilde over (w)})}_(i)(t)=−γ((x _(i)(t)−{circumflex over (x)}_(i)(t)+2μ(ŵ _(i)(t)−x _(i)(t)))−ċ(t), {tilde over (w)} _(i)(0)=w_(i0)   Equation 19

where {tilde over (w)}_(i)(0)=ŵ_(i)(0)−c(0).

Equation 20 and 21 are defined below to provide background informationregarding a first theorem. Equation 20 is described as:

$\begin{matrix}{{v\left( {{e(t)},{\overset{\sim}{w}(t)}} \right)}\overset{\bigtriangleup}{=}{{\frac{1}{2}{e^{T}(t)}{F^{- 1}(G)}{e(t)}} + {\frac{1}{2}{{\overset{\sim}{w}}^{T}(t)}\Gamma^{- 1}{\overset{\sim}{w}(t)}}}} & {{Equation}20}\end{matrix}$

where

${\overset{\sim}{w}(t)}\overset{\bigtriangleup}{=}{{\left\lbrack {{{\overset{\sim}{w}}_{1}(t)},\ldots,{{\overset{\sim}{w}}_{n}(t)}} \right\rbrack^{T} \in {{\mathbb{R}}^{n}{and}\Gamma}}\overset{\bigtriangleup}{=}{{{diag}\left( \left\lbrack {\gamma_{1},\ldots,\gamma_{n}} \right\rbrack^{T} \right)}.}}$

Equation 21 is described as:

$\begin{matrix}{a\overset{\bigtriangleup}{=}{\min\left\{ {{2a_{1}/{\overset{¯}{\lambda}\left( {F^{- 1}(G)} \right)}},{2a_{2}/{\overset{¯}{\lambda}\left( \Gamma^{- 1} \right)}}} \right\}}} & {{Equation}21}\end{matrix}$

where we define

${{a1}\overset{\bigtriangleup}{=}{1 - \mu - {1/\left( {2h_{1}} \right)}}},$

h₁∈

₊ being an arbitrary to ensure a₁∈

₊ and

${{a2}\overset{\bigtriangleup}{=}{\mu - {1/\left( {2h_{2}} \right)}}},$

h₂∈

₊ being an arbitrary to ensure a₂∈

₊. The time-varying command c(t) is bounded by time rate of change andsatisfies both ∥F⁻¹(G)ρ(t)∥₂ ²≤≤ρ₁ and ∥ρ(t)∥₂ ²≤ρ₂, where ρ₁∈

₊, ρ₂∈

₊, and

$\rho_{0}\overset{\bigtriangleup}{=}{{\rho_{1}\Delta/\left( {2h_{1}} \right)} + {\rho_{2}/{\left( {2h_{1}} \right).}}}$

The first theorem is directed to a multiagent system consisting of aplurality of agent denoted as n, over a connected and undirected graphG. The distributed control architecture defined by equation 11 with thenorm-free and adaptive event-triggering rule provided by equation 12 and13. The solution of the close-loop multiagent system may be describedas:

$\begin{matrix}{{v\left( {{e(t)},{\overset{˜}{w}(t)}} \right)} \leq {{{v\left( {{e(0)},{\overset{˜}{w}(0)}} \right)}e^{{- \alpha}t}} + \frac{\rho_{0}}{\alpha}}} & {{Equation}22}\end{matrix}$

The solution of the closed-loop multiagent system described in equation22 is globally exponentially stable when c(t) is constant as isdescribed as:

$\begin{matrix}{{\lim\limits_{t\rightarrow\infty}\left( {{e(t)},{\overset{˜}{w}(t)}} \right)} = \left( {0,0} \right)} & {{Equation}23}\end{matrix}$

The first portion of the equation 20 may be expressed as below:

$\begin{matrix}{{v_{1}\left( {e(t)} \right)}\overset{\bigtriangleup}{=}{\frac{1}{2}{e^{T}(t)}{F^{- 1}(G)}{e(t)}}} & {{Equation}24}\end{matrix}$

The derivative of equation 24 with respect to time is described below:

{dot over (v)} ₁(e(t))=−∥e(t)∥₂ ² +e ^(T)(t)(x(t)−{circumflex over(x)}(t)=−e ^(T)(t)F ⁻¹(G)ρ(t)  Equation 25

The event-triggering rule is described as

e _(i)(t)((x _(i)(t)−{circumflex over (x)} _(i)(t))=(x _(i)(t)−ŵ_(i)(t))(x _(i)(t)−{circumflex over (x)} _(i)(t))+{tilde over (w)}_(i)(x _(i)(t)−{circumflex over (x)} _(i)(t))   Equation 26

The event-triggering rule expressed by equation 12 and 16 can be upperbounded as:

e _(i)(t)((x _(i)(t)−{circumflex over (x)} _(i)(t))≤μ(x _(i)(t)−ŵ_(i)(t))² +{tilde over (w)} _(i)(x _(i)(t)−{circumflex over (x)}_(i)(t))  Equation 27

The right-hand side of the equation μ(x_(i)(t)−ŵ_(i)(t)))² can bedescribed as:

μ(x _(i)(t)−ŵ _(i)(t))²=μ({tilde over (w)} _(i)(t)−e _(i)(t))²  Equation28

Incorporating equation 26 with 27, equation 25 can be rewritten as:

$\begin{matrix}{{{{\overset{.}{v}}_{1}\left( {e(t)} \right)} \leq {{- {{e(t)}}_{2}^{2}} + {\sum\limits_{i = 1}^{n}{{\overset{˜}{w}}_{i}\left( {\left( {{x_{i}(t)} - {{\overset{\hat{}}{x}}_{i}(t)}} \right) + {\mu\left( {{{\overset{˜}{w}}_{i}(t)} - {2{e_{i}(t)}}} \right)}} \right)}} + {\sum\limits_{i = 1}^{n}{\mu{e_{i}^{2}(t)}}} - {{e^{T}(t)}{F^{- 1}(G)}{\rho(t)}}}} = {{{- \left( {1 - \mu} \right)}{{e(t)}}_{2}^{2}} + {\sum\limits_{i = 1}^{n}{{{\overset{˜}{w}}_{i}(t)}\left( {\left( {{x_{i}(t)}\  - \ {{\overset{\hat{}}{x}}_{i}(t)}} \right) + {\mu\left( {{{\overset{\hat{}}{w}}_{i}(t)}\  - \ {x_{i}(t)}} \right)}} \right)}} - {\sum\limits_{i = 1}^{n}{\mu{{\overset{˜}{w}}_{i}(t)}{e_{i}(t)}}} - {{e^{T}(t)}{F^{- 1}(G)}{\rho(t)}}}} & {{Equation}29}\end{matrix}$

where the term −μ{tilde over (w)}_(i)(t)e_(i)(t) may be described as:

−μ{tilde over (w)} _(i)(t)e _(i)(t)=−μ{tilde over (w)} _(i)(t)(x_(i)(t)−ŵ _(i)(t))−−μ{tilde over (w)} _(i) ²(t)  Equation 30

Incorporating equation 29 and 30, equation 29 may be rewritten as:

$\begin{matrix}{{{\overset{.}{v}}_{1}\left( {e(t)} \right)} \leq {{{- \left( {1 - \mu} \right)}{{e(t)}}_{2}^{2}} - {\mu{{\overset{˜}{w}(t)}}_{2}^{2}} + {\sum\limits_{i = 1}^{n}{{{\overset{˜}{w}}_{i}(t)}\left( {\left( {{x_{i}(t)} - {{\overset{\hat{}}{x}}_{i}(t)}} \right) + {2{\mu\left( {{{\overset{\hat{}}{w}}_{i}(t)} - {x_{i}(t)}} \right)}}} \right)}} - {{e^{T}(t)}{F^{- 1}(G)}{\rho(t)}}}} & {{Equation}31}\end{matrix}$

Further, the second portion of the equation 20 may be expressed asbelow:

v ₂({tilde over (w)}(t))=½{tilde over (w)} ^(T)(t)Γ⁻¹ {tilde over(w)}(t)  Equation 32

Incorporating the first portion and the second portion v(e(t),{tildeover (w)}(t))=v₁(e(t))+v₂ ({tilde over (w)}(t)), a Lyapunov-likefunction is provided similar to equation 20. With respect to the firstlemma provided above, v(e(t),{tilde over (w)}(t))∈

₊ as v(e(t),{tilde over (w)}(t)) is radially unbounded. By taking aderivative of equation 20 with respect to time:

{dot over (v)}(e(t),{tilde over (w)}(t))≤−(1−μ)∥e(t)∥₂ ² −μ∥{tilde over(w)}(t)∥₂ ² −e ^(T)(t)F ⁻¹(G)ρ(t)−{tilde over (w)} ^(T)(t)Γ⁻¹ρ(t)  Equation 33

Incorporating the second lemma for −e^(T)(t)F⁻¹(G)ρ(t)−{tilde over(w)}^(T)(t)Γ⁻¹ρ(t):

{dot over (v)}(e(t),{tilde over (w)}(t))≤−a ₁ ∥e(t)∥₂ ² −a ₂ ∥{tildeover (w)}(t)∥₂ ²+ρ₀ ,−av(e(t),{tilde over (w)}(t))+ρ₀   Equation 34

By incorporating the third lemma, equation 22 may be derived. Fortime-varying command c(t) being a constant, ρ₀=0 in equation 34 and thesolution (e(t),{tilde over (w)}(t)) of the closed-loop multiagent systemis globally exponentially stable as depicted in equation 23.

The event-triggering rule given by equation 25 for sampled datainformation exchange or information exchange predicated on the solutionpredictor curve method for may be different from equation 12 and 13 andthe global information exchange is expressed as below:

e _(i)(t)(x _(i)(t)−{circumflex over (x)} _(i)(t))≤μe _(i)²(t)  Equation 35

Equation 35 is dependent on e_(i)(t)=x_(i)(t)−c(t) which provides thecommand that is available only to the leader agents and is unavailableto other agents unless the global information exchange is allowed. Itfollows that:

{dot over (v)} ₁(e(t))≤=−(1−μ)∥e(t)∥₂ ² −e ^(T)(t)F ⁻¹(G)ρ(t)  Equation36

where the first theorem and the second and third lemmas define uniformultimate boundedness when the command is time varying and is globallyexponentially stable when the command is constant without the need foran adaptive term.

The global information exchange is not scalable due to significantlyincreased communication cost, not secure due to exchanging informationgoals of all agents, and not feasible for cases involving large numbersof agents. In order to mitigate deficiencies listed above, adaptiveestimate ŵ_(i)(t) of c(t) is first added an then subtracted from theterm e_(i)(t)(x_(i)(t)−{circumflex over (x)}_(i)(t)) of equation 26 andis described in the first theorem. The introduction of the adaptive termin the event-triggering rule for each agent as provided by equation 12and 13, both decentralized and norm-free features is enabledsimultaneously without relying on any global information.

Referring to FIG. 1 , the event triggering approach with four agents ona connected and undirected graph with the first agent being a leaderagent and having the knowledge of the command is provided as:

$\begin{matrix}{{c(t)} = \left\{ \begin{matrix}A & {{{{for}\ 0} \leq t \leq 25},} \\{A + B} & {\ {{{{for}\ 25} \leq t \leq {50}},}} \\{A - B} & {\ {{{{for}\ 50} \leq t \leq {75}},}}\end{matrix} \right.} & {{Equation}40}\end{matrix}$

Where A=(x₁₀+x₂₀+x₃₀+x₄₀)/4 and B=1.2. The initial condition of theagent is set to be x₁₀=3.2, x₂₀=2.3, x₃₀=1.4, x₄₀=0.5 and μ=0.95 for theevent-triggering rule provided by equation 12. The speed or rate ofconvergence of the adaptive term provided by equation 13 includesγ_(i)=0.85 and γ_(i)=2.5 for all agents. The sampling time is set to0.05 seconds that yields to 6000 data points being exchanged without theevent-triggering approach.

FIGS. 2 and 3 illustrates the closed-loop multiagent system 200, 300response when rate of convergence of the adaptive term provided byequation 13 is set to γ_(i)=0.85. The first agent 204, 304 isrepresented by A1+, which is a leader agent, the second agent, 208, 308is represented by A2, the third agent 212, 312 is represented by A3, andthe fourth agent 216, 316 is represented by A4. A time varying command230, 330 is illustrated as dotted lines. FIGS. 2 and 3 includes a firstplot 240, 340, a second plot 244, 344, a third plot 248, 348, and afourth plot 252, 352. The first plot 240, 340 includes the relationshipof x(t), which represents a state (e.g., a current state) of the agent,and time (in second). The second plot 244,344 includes the relationshipof OW (an adaptive term of the agent) and time (in seconds). The thirdplot 248,348 includes the relationship of u(t), which represents acontrol signal of the agent), and time (in seconds). The fourth plot252,352, includes the relationship of number of events of the firstagent 204, 304, the second agent 208, 308, the third agent 212, 312, andfourth agent 216, 316 over a period of time (in seconds).

FIGS. 4 and 5 illustrates the closed loop multiagent system 400, 500response when rate of convergence of the adaptive term provided byequation 13 is set to γ_(i)=2.5. The first agent 404, 504 is representedby A1+, which is a leader agent, the second agent, 408, 508 isrepresented by A2, the third agent 412, 512 is represented by A3, andthe fourth agent 416, 516 is represented by A4. A time varying command430, 530 is illustrated as dotted lines. FIGS. 4 and 5 includes a firstplot 440, 540, a second plot 444, 544, a third plot 448, 548, and afourth plot 452, 552. The first plot 440, 540 includes the relationshipof x(t) and time (in second). The second plot 444,544 includes therelationship of ŵ(t) and time (in seconds). The third plot 448,548includes the relationship of u(t) and time (in seconds). The fourth plot452,552, includes the relationship of number of events of the firstagent 404, 504, the second agent 408, 508, the third agent 412, 512, andfourth agent 416, 516 over a period of time (in seconds).

The lower γ_(i) provides fewer number of events predicated on both thesampled data exchange approach and the solution-predictor curveapproach. The lower number of events during the exchange of sampled datapoints, the solution-predictor method is less effective. Referring toFIGS. 4 and 5 in comparison to FIGS. 2 and 3 for sampled data exchangeand solution predictor curve exchange, the system response of theclosed-loop multiagent system degrades when using lower values of γ_(i).

The response of the closed-loop multiagent system response improves asthe number of events or the y value is increased. The system response ofthe closed-loop multiagent system degrades when using lower values ofγ_(i). increases. Referring to FIGS. 4 and 5 , the solution-predictormethod becomes significantly effective when number of events increasesusing sampled data exchange.

Prior systems utilize only the solution-predictor method but do notappreciate or suggest the norm-free and/or the adaptive approach of thepresently proposed event-triggering rule. The solution-predictor methodwithout the norm-free and adaptive approach reduced the number of eventsto 1837 when sampled data exchange is used and the number of events wasreduced to 1285 for solution-predictor curve. Referring to FIG. 4 , thenumber of events reduced to 998 events when sampled data exchange isused and referring to FIG. 5 , the number of events reduced to 487events when solution-predictor curve is used. This comparison shows thatthe presently proposed event-triggering rule with norm-free and adaptiveapproach provides advantage in further reducing agent-to-agentinformation exchange. The reduction of agent-to-agent informationexchange is enabled due to φ(t)=0. The event triggering rule of priorart is defined by ∥μ _(i)(t)∥₂≤ε∥ē_(i)(t)∥₂+φ(t) where φ(t)∈

is an exponentially decaying term. The presently proposedevent-triggering rule does not include a similar term, and φ(t) may beset to zero.

Reducing agent-to-agent information exchange in distributed control ofmultiagent system is enabled by a new event-triggering rule. The newevent triggering rule is subject to sampled data information exchange orinformation exchange predicated on the solution-predictor curve method.The decentralized and norm-free feature of the new event-trigger ruleincludes the adaptive term. The adaptive term utilizes each agent toestimate unknown variables unavailable to a designated agent. Based onthe first, second, third lemmas, system-theoretical analysis illustratesthat the solution-predictor curve method has the ability to furtherreduce agent-to-agent information exchange, where each agent stores thecurve and exchanges the parameters when an event occurs in a distributedmanner for approximating the solution trajectory of each agent.

FIG. 6 is a block diagram conceptually illustrating a multiagent systemfor information exchange. For example, the multiagent system 600 caninclude one or more agents (i.e., follower agents) 610, 610 n and one ormore leader agents 630, 630 n. In some examples, the one or more agents610, 610 n and the one or more leader agents 630, 630 n can be connectedvia a communication network 650 with a connected and undirected graph.

In some examples, the agent 610, 610 n can transmit or receiveinformation (e.g., the status of the agent 610, 610 n) over acommunication network 650. In some examples, the communication network650 can be any suitable communication network or combination ofcommunication networks. For example, the communication network 650 caninclude a Wi-Fi network (which can include one or more wireless routers,one or more switches, etc.), a peer-to-peer network (e.g., a Bluetoothnetwork), a cellular network (e.g., a 3G network, a 4G network, a 5Gnetwork, etc., complying with any suitable standard, such as CDMA, GSM,LTE, LTE Advanced, NR, etc.), a wired network, etc. In some embodiments,communication network 650 can be a local area network, a wide areanetwork, a public network (e.g., the Internet), a private orsemi-private network (e.g., a corporate or university intranet), anyother suitable type of network, or any suitable combination of networks.Communications links between agents 610, 610 n, between agents 610, 610n and leader agents 630, 630 n, and/or between leader agents 630, 630 ncan each be any suitable communications link or combination ofcommunications links, such as wired links, fiber optic links, Wi-Filinks, Bluetooth links, cellular links, etc.

In further examples, the agent 610, 610 n and/or a leader agent 630, 630n can be a vehicle (e.g., a ground vehicle, an aerial vehicle, anunderwater vehicle, a ship, a space vehicle, an autonomous vehicle, amotor vehicle, a space vehicle, car, a train, an unmanned aerialvehicle, a rocket, or a missile, etc.) or any other suitable apparatusor means (e.g., a computing/circuit/electrical node). The agent 610, 610n can include any suitable computing device or combination of devices,such as a processor (including an ASIC, DSP, PFGA, or other processingcomponent) desktop computer, a laptop computer, a smartphone, a tabletcomputer, a wearable computer, a server computer, a computing deviceintegrated into the vehicle, a camera, a robot, a virtual machine beingexecuted by a physical computing device, etc.

In further examples, the agent 610, 610 n can include a processor 612, adisplay 614, one or more inputs 616, one or more communication systems618, and/or memory 620. In some embodiments, the processor 612 can beany suitable hardware processor or combination of processors, such as acentral processing unit (CPU), a graphics processing unit (GPU), anapplication specific integrated circuit (ASIC), a field-programmablegate array (FPGA), a digital signal processor (DSP), a microcontroller(MCU), etc. In some embodiments, the display 614 can include anysuitable display devices, such as a computer monitor, a touchscreen, atelevision, an infotainment screen, etc. In some embodiments, theinput(s) 616 can include any suitable input devices and/or sensors thatcan be used to receive user input, such as a keyboard, a mouse, atouchscreen, a microphone, etc.

In further examples, the communications system 618 can include anysuitable hardware, firmware, and/or software for communicatinginformation over communication network 650 and/or any other suitablecommunication networks. For example, the communications system 618 caninclude one or more transceivers, one or more communication chips and/orchip sets, etc. In a more particular example, the communications system618 can include hardware, firmware and/or software that can be used toestablish a Wi-Fi connection, a Bluetooth connection, a cellularconnection, an Ethernet connection, etc. to transmit the status of theagent 610, 610 n and/or receive the status of the one or moreneighboring agents 610, 610 n, 630, 630 n.

In further examples, the memory 620 can include any suitable storagedevice or devices that can be used to store status of the agent 610, 610n, status of the one or more neighboring agents 610, 610 n, 630, 630 n,a solution predictor curve of the agent 610, 610 n, one or more solutionpredictor curves of the one or more neighboring agents 610, 610 n, 630,630 n, data, instructions, values, etc., that can be used, for example,by the processor 612 to perform information exchange tasks viacommunications system 618, etc. The memory 620 can include any suitablevolatile memory, non-volatile memory, storage, or any suitablecombination thereof. For example, memory 610 can include random accessmemory (RAM), read-only memory (ROM), electronically-erasableprogrammable read-only memory (EEPROM), one or more flash drives, one ormore hard disks, one or more solid state drives, one or more opticaldrives, etc. In some embodiments, the memory 620 can have encodedthereon a computer program for controlling operation of computing device610. For example, in such embodiments, the processor 612 can execute atleast a portion of the computer program to perform one or more dataprocessing tasks described herein, transmit/receive information via thecommunications system 618, etc. As another example, processor 612 canexecute at least a portion of process 700 described below in connectionwith FIG. 7 .

In even further examples, the multiagent system can include one or moreleader agents 630, 630 n. A leader agent 630, 630 n can include aprocessor 632, a display 634, input(s) 636, a communication system 638,and/or a memory 640. In some examples, the processor 632, the display634, the input(s) 636, the communication system 638, and/or the memory640 of the leader 630, 630 n are substantially similar to those in theagent 610, 610 n. In addition, the leader 630, 630 n can receive abounded time-varying command (c(t)) which the one or more agents 610,610 n and the one or more leader agents 630, 630 n are to follow. Insome examples, the one or more agents 610, 610 n may not access or knowthe command (c(t)). In other examples, the command (c(t)) can beavailable to the one or more agents 610, 610 n.

FIG. 7 is a flowchart illustrating an example process for informationexchange. In some examples, the process 700 for information exchange ina multiagent system may be carried out by the agent 610, 610 n and/orthe leader agent 630 illustrated in FIG. 6 . In some examples, themultiagent system can include multiple homogeneous or heterogeneousagents to communicate each other. An agent 610, 610 n, 630 in themultiagent system can include a vehicle (e.g., a ground vehicle, anaerial vehicle, an underwater vehicle, a ship, a space vehicle, anautonomous vehicle, a motor vehicle, a space vehicle, car, a train, anunmanned aerial vehicle, a rocket, or a missile, etc.) operating in agroup or team. However, it should be appreciated that the agent 610, 610n, 630 can be any other suitable apparatus or means (e.g., acomputing/circuit/electrical node, a robot, a power node in a powergrid) for carrying out the functions or algorithm described below.Additionally, although the steps of the flowchart 700 are presented in asequential manner, in some examples, one or more of the steps may beperformed in a different order than presented, in parallel with anotherstep, or bypassed.

At step 702, the process for information exchange of a vehicle in amultiagent system can receive one or more neighboring states broadcastby one or more neighboring vehicles. In some examples, the one or moreneighboring states can correspond to the one or more neighboringvehicles. In some examples, a neighboring vehicle (j) can transmit aneighboring state ({circumflex over (x)}_(j)(t)) when the neighboringstate ({circumflex over (x)}_(j)(t)) is an initial state ({circumflexover (x)}_(j)(0) or ({circumflex over (x)}_(j0))) or the neighboringstate ({circumflex over (x)}_(j)(t)) causes violation of an informationexchange triggering condition (which is elaborated below in connectionwith step 708 in detail). Then, the vehicle (i) can receive theneighboring state ({circumflex over (x)}_(j)(t)) broadcasted by theneighboring state.

At step 704, the process can transmit a last broadcast state({circumflex over (x)}_(i)(t)) of the vehicle to the one or moreneighboring vehicles (j). Each vehicle (i) can transmit its initialstate (x_(i)(0) or x_(i0)) to its neighboring vehicle(s) (j) or itsstate x_(i)(t) to its neighboring vehicles (j) when the state (x_(i)(t))causes violation of an information exchange triggering condition. Thestate of a vehicle and the information exchange triggering condition areexplained further in connection with step 706 and 708, respectivelybelow. In some examples, the last broadcast state ({circumflex over(x)}_(i)(t)) can be the latest state broadcast or transmitted to the oneor more neighboring vehicles (j). In some examples, the last broadcaststate ({circumflex over (x)}_(i)(t)) can be a constant real valuebecause the last broadcast state can be a state transmitted previously.For example, the last broadcast state ({circumflex over (x)}_(i)(t)) ofthe vehicle (i) at time t can be a state at time t−2 (i.e., (x_(i)(t−2))because the most recent or latest state of the vehicle (i) transmittedor broadcast is the state at time t−2. In the example, the lastbroadcast state ({circumflex over (x)}_(i)(t)) of the vehicle (i) attime t can be a constant value at time t−2 and t−1. In other examples,the last broadcast state can be a time-varying value (e.g., based on asolution-predictor curve).

At step 706, the process can determine a current state (x_(i)(t)) of thevehicle (i) based on the one or more neighboring states ({circumflexover (x)}_(j)(t)) and the last broadcast state ({circumflex over(x)}_(i)(t)). In some examples, a state of a vehicle can indicate alocation (e.g., Global Positioning System (GPS) position, (x, y)position, (x, y, z) position) of the vehicle, a phase angle in a powersystem, or an output or a result of a control signal (u_(i)(t)) of thevehicle. For example, when there are n vehicles (i=1, . . . , n), thestate of each agent can be defined as: x_(i)(t)=u_(i)(t), where theinitial state (x_(i)(0)) of each vehicle can be x_(w). Here, the currentstate (x_(i)(t) of the vehicle can be a time-varying real number orvalue, and u_(i)(t) is a control signal. In some examples, the controlsignal u_(i)(t) can be defined as: u_(i)(t)=−Σ_(i˜j)({circumflex over(x)}_(i)(t)−{circumflex over (x)}_(j)(t))−k_(i)({circumflex over(x)}_(i)(t)−c(t)), where {circumflex over (x)}_(i)(t) is the lastbroadcast state of the vehicle (i) at time t, {circumflex over(x)}_(j)(t) is the one or more neighboring states at time t, k_(i)=1 forleader vehicles, k_(i)=0 for follower vehicles, and c(t) is a boundedtime-varying command that has piecewise continuous and bounded time rateof change. In some examples, the command is only available to a subsetof vehicles (i.e., leader vehicles), and the current states of allvehicles are driven to this command. In a non-limiting scenario, thecommand can be a certain trajectory (e.g., a path between two points) ora direction to move the vehicles to a specific place. However, it shouldbe appreciated that the command can be any other suitable task formultiple vehicles to perform as an objective (e.g., global objective) ofthe controlled multiagent system. In further examples, the command isunknown and unavailable to a follower vehicle such that the controlsignal u_(i)(t) is determined by u_(i)(t)=−Σ_(i˜j)({circumflex over(x)}_(i)(t)−{circumflex over (x)}_(j)(t)).

At step 708, the process can determine a norm-free information exchangetriggering condition based on the last broadcast state, the currentstate, and an estimated command. The norm-free information exchangetriggering condition can be referred to a norm-free and adaptiveevent-triggering rule. In some examples, the estimated command caninclude an estimate of the command such that the current state is drivento follow the command although the command is unavailable and unknown tothe vehicle. In further examples, the norm-free information exchangetriggering condition can depend on error signals of the vehicle withoutusing neighboring information or global information (e.g., the command).For example, the norm-free information exchange triggering condition canbe defined by: (x_(i)(t)−ŵ_(i)(t))(x_(i)(t)−{circumflex over(x)}_(i)(t))≤μ((x_(i)(t)−ŵ_(i)(t))², where x_(i)(t) is the current stateof the vehicle at time t, ŵ_(i)(t) is the estimated command or adaptiveterm, {circumflex over (x)}_(i)(t) is the last broadcast state of thevehicle, and μ is an error scaling parameter. Contrary to conventionevent-triggering rules in a multiagent system, a negative value of theleft side (i.e., (x_(i)(t)−ŵ_(i)(t))(x_(i)(t)−{circumflex over(x)}_(i)(t))) in the norm-free information exchange triggering conditioncan avoid violation of the norm-free information exchange triggeringcondition. That is, when the left side of the norm-free informationexchange triggering condition is negative, it is not possible to violatethe norm-free information exchange triggering condition because theright side (i.e., μ((x_(i)(t)−ŵ_(i)(t))²) of the norm-free informationexchange triggering condition has only positive values. This can implybetter information exchange reduction because the only condition toviolate the norm-free information exchange triggering condition is thatthe left side of the norm-free information exchange triggering conditionis positive and larger that the right side of the norm-free informationexchange triggering condition.

In some examples, the estimated command can be defined asŵ_(i)(t)=−γ_(i)((x_(i)(t)−{circumflex over(x)}_(i)(t))+2μ(ŵ_(i)(t)−x_(i)(t))), where the initial estimated command(i.e., ŵ_(i)(0)) is ŵ_(i0), x_(i)(t) is the current state of the vehicleat time t, {circumflex over (x)}_(i)(t) is the last broadcast state ofthe vehicle, μ is the error scaling parameter, and γ_(i) is an adaptiveparameter of the vehicle.

At step 710, the process can transmit the current state to the one ormore neighboring vehicles in response to the current state violating thenorm-free information exchange triggering condition. In some examples,as the current state of the vehicle changes in time domain, the changingcurrent state can determine whether the norm-free information exchangetriggering condition is violated. In a non-limiting scenario, thenorm-free information exchange triggering condition can be violated whenthe vehicle moves to an opposite or different direction to or from thedirection that the command directs. For example, at time p, the currentstate can result in (x_(i)(p)−ŵ_(i)(p))(x_(i)(p)−{circumflex over(x)}_(i)(p))>μ((x_(i)(p)−ŵ_(i)(p))² such that the left side of thenorm-free information exchange triggering condition is greater than theright side of the norm-free information exchange triggering condition.Then, the norm-free information exchange triggering condition isviolated, and the vehicle can transmit the current state (x_(i)(p)) attime p to the one or more neighboring vehicles.

In some scenarios, to transmit the current state to the one or moreneighboring vehicles, the process can transmit a sampled dataset of thecurrent state through a zero-order-hold operator to the one or moreneighboring vehicles. In some examples, the scenarios depend on thezero-order-hold operator to reconstruct a signal using adigital-to-analog converter (DAC).

In other scenarios, the process can a use solution-predictor curvemethod to further reduce vehicle-to-vehicle information exchange. Forexample, the process can further transmit a solution-predictor curve ofthe current state to the one or more neighboring vehicles forapproximating a trajectory of the vehicle in response to the currentstate violating the norm-free information exchange triggering condition.In some examples, the solution-predictor curve of the current state canbe defined as:

${{\xi_{i}(t)}\overset{\bigtriangleup}{=}{{{\exp\left( {Q_{i}\left( {t - t_{i}^{e}} \right)} \right)}{x_{i}\left( t_{i}^{e} \right)}} + {\frac{B_{i}}{Q_{i}}\left( {1 - {\exp\left( {Q_{i}\left( {t - t_{i}^{e}} \right)} \right)}} \right)}}},{{{where}B_{i}}\overset{\bigtriangleup}{=}{{\sum_{i\sim j}{\overset{\hat{}}{x}}_{j}} + {k_{i}r}}},{Q_{i} = {k_{i} + d_{i}}},$

determines whether an agent is leader or follower, denotes the number ofneighbors of an agent,

${{{\overset{\hat{}}{x}}_{i}(t)}\overset{\bigtriangleup}{=}{x_{i}(t)}},{{{and}{\overset{\hat{}}{x}}_{j}}\overset{\bigtriangleup}{=}{x_{j}\left( t_{j}^{e} \right)}},{{{and}r}\overset{\bigtriangleup}{=}{{c\left( t_{i}^{e} \right)}.}}$

In further examples, the one or more neighboring states can include oneor more solution-predictor curves corresponding to the one or moreneighboring vehicles and uses ξ_(j)(t) as one or more neighboring statesbroadcast by one or more neighboring vehicles. In even further examples,the process can use ξ_(i)(t) as the last broadcast state broadcast toone or more neighboring vehicles. Thus, the one or more neighboringstates and the last broadcast state can be time-varying real values.

In the foregoing specification, implementations of the disclosure havebeen described with reference to specific example implementationsthereof. It will be evident that various modifications may be madethereto without departing from the broader spirit and scope ofimplementations of the disclosure as set forth in the following claims.The specification and drawings are, accordingly, to be regarded in anillustrative sense rather than a restrictive sense.

What is claimed is:
 1. A method for information exchange of a vehicle ina multiagent system, the method comprising: receiving one or moreneighboring states broadcast according to an undirected and connectedgraph topology by one or more neighboring vehicles; transmitting a lastbroadcast state of the vehicle to the one or more neighboring vehicles;determining a current state of the vehicle based on the one or moreneighboring states and the last broadcast state; determining a norm-freeinformation exchange triggering condition based on the last broadcaststate, the current state, and an estimated command; and in response tothe current state violating the norm-free information exchangetriggering condition, transmitting the current state to the one or moreneighboring vehicles.
 2. The method of claim 1, wherein the lastbroadcast state comprises a constant real value through thezero-order-hold operator or through time-varying real value throughsolution predictor curve method, and wherein the current state comprisesa time-varying real value.
 3. The method of claim 1, wherein the lastbroadcast state is a state of the vehicle most currently transmitted tothe one or more neighboring vehicles.
 4. The method of claim 1, whereinthe current state of the vehicle comprises a control signal of thevehicle.
 5. The method of claim 4, wherein the control signal isdetermined by: u_(i)(t)=−Σ_(i˜j)({circumflex over(x)}_(i)(t)−{circumflex over (x)}_(j)(t)), where u_(i)(t) is the controlsignal of the vehicle at time t, {circumflex over (x)}_(i)(t) is thelast broadcast state of the vehicle at time t, and {circumflex over(x)}_(j)(t) is the one or more neighboring states.
 6. The method ofclaim 1, wherein the estimated command is an estimate of a command suchthat the current state is driven to follow the command.
 7. The method ofclaim 6, wherein the command is unknown and unavailable to the vehiclesentitled followers.
 8. The method of claim 1, wherein the norm-freeinformation exchange triggering condition depends on error signals ofthe vehicle without using neighboring information or global information.9. The method of claim 1, wherein a negative value of a left side in thenorm-free information exchange triggering condition avoids violation ofthe norm-free information exchange triggering condition.
 10. The methodof claim 1, wherein the norm-free information exchange triggeringcondition is defined as: (x_(i)(t)−ŵ_(i)(t))(x_(i)(t)−{circumflex over(x)}_(i)(t))≤μ((x_(i)(t)−ŵ_(i)(t))², where x_(i)(t) is the current stateat time t, ŵ_(i)(t) is the estimated command, {circumflex over(x)}_(i)(t) is the last broadcast state, and μ is an error scalingparameter.
 11. The method of claim 10, wherein the estimated command isdefined as: ŵ_(i)(t)=−γ_(i)((x_(i)(t)−{circumflex over(x)}_(i)(t))+2μ(ŵ_(i)(t)−x_(i)(t))), where x_(i)(t) is the current stateof the vehicle at time t, {circumflex over (x)}_(i)(t) is the lastbroadcast state of the vehicle, μ is an error scaling parameter, andγ_(i) is an adaptive parameter of the vehicle.
 12. The method of claim1, wherein transmitting the current state to the one or more neighboringvehicles comprises: transmitting a sampled dataset of the current statethrough a zero-order-hold operator to the one or more neighboringvehicles.
 13. The method of claim 1, further comprising: in response tothe current state violating the norm-free information exchangetriggering condition, transmitting a solution-predictor curve of thecurrent state to the one or more neighboring vehicles for approximatinga trajectory of the vehicle.
 14. The method of claim 13, wherein thesolution-predictor curve of the current state is defined as:${{\xi(t)}\overset{\bigtriangleup}{=}{{{\exp\left( {Q_{i}\left( {t - t_{i}^{e}} \right)} \right)}{x_{i}\left( t_{i}^{e} \right)}} + {\frac{B_{i}}{Q_{i}}\left( {1 - {\exp\left( {Q_{i}\left( {t - t_{i}^{e}} \right)} \right)}} \right)}}},{{{where}B_{i}}\overset{\bigtriangleup}{=}{{\sum_{i\sim j}{\overset{\hat{}}{x}}_{j}} + {k_{i}r}}},{Q_{i} = {k_{i} + d_{i}}},t_{i}^{e}$is a time when the norm-free information exchange triggering conditionis violated.
 15. The method of claim 1, wherein the one or moreneighboring states comprises one or more solution-predictor curvescorresponding to the one or more neighboring vehicles.
 16. The method ofclaim 1, wherein the last broadcast state is a time-varying real value.17. An agent for information, the agent comprising: a processor; and amemory having stored thereon a set of instructions which, when executedby the processor, cause the processor to: receive one or moreneighboring states broadcast by one or more neighboring vehicles;transmit a last broadcast state of the agent to the one or moreneighboring vehicles; determine a current state of the vehicle based onthe one or more neighboring states and the last broadcast state;determine a norm-free information exchange triggering condition based onthe last broadcast state, the current state, and an estimated command;and in response to the current state violating the norm-free informationexchange triggering condition, transmit the current state to the one ormore neighboring vehicles.
 18. The agent of claim 17, wherein the lastbroadcast state comprises a constant real value through thezero-order-hold operator or through time-varying real value throughsolution predictor curver method, and wherein the current statecomprises a time-varying real value.
 19. The agent of claim 17, whereinthe last broadcast state is a state of the vehicle most currentlytransmitted to the one or more neighboring vehicles.
 20. The agent ofclaim 17, wherein the current state of the vehicle comprises a controlsignal of the vehicle.
 21. The agent of claim 20, wherein the controlsignal is determined by: u_(i)(t)=−Σ_(i˜j)({circumflex over(x)}_(i)(t)−{circumflex over (x)}_(j)(t)), where u_(i)(t) is the controlsignal of the vehicle at time t, {circumflex over (x)}_(i)(t) is thelast broadcast state of the vehicle at time t, and {circumflex over(x)}_(j)(t) is the one or more neighboring states.
 22. The agent ofclaim 17, wherein the estimated command is an estimate of a command suchthat the current state is driven to follow the command.
 23. The agent ofclaim 22, wherein the command is unknown and unavailable to the vehiclesentitled followers.
 24. The agent of claim 17, wherein the norm-freeinformation exchange triggering condition depends on error signals ofthe vehicle without using neighboring information or global information.25. The agent of claim 17, wherein a negative value of a left side inthe norm-free information exchange triggering condition avoids violationof the norm-free information exchange triggering condition.
 26. Theagent of claim 17, wherein the norm-free information exchange triggeringcondition is defined as: (x_(i)(t)−ŵ_(i)(t))(x_(i)(t)−{circumflex over(x)}_(i)(t))≤μ((x_(i)(t)−ŵ_(i)(t))², where x_(i)(t) is the current stateat time t, ŵ_(i)(t) is the estimated command, {circumflex over(x)}_(i)(t) is the last broadcast state, and μ is an error scalingparameter.
 27. The agent of claim 26, wherein the estimated command isdefined as: ŵ_(i)(t)=−γ_(i)((x_(i)(t)−{circumflex over(x)}_(i)(t))+2μ(ŵ_(i)(t)−x_(i)(t))), where x_(i)(t) is the current stateof the vehicle at time t, {circumflex over (x)}_(i)(t) is the lastbroadcast state of the vehicle, μ is an error scaling parameter, andγ_(i) is an adaptive parameter of the vehicle.
 28. The agent of claim17, wherein transmitting the current state to the one or moreneighboring vehicles comprises: transmitting a sampled dataset of thecurrent state through a zero-order-hold operator to the one or moreneighboring vehicles.
 29. The agent of claim 17, wherein the memoryhaving stored thereon a further set of instructions which, when executedby the processor, cause the processor to: in response to the currentstate violating the norm-free information exchange triggering condition,transmit a solution-predictor curve of the current state to the one ormore neighboring vehicles for approximating a trajectory of the vehicle.30. The agent of claim 29, wherein the solution-predictor curve of thecurrent state is defined as:${{\xi(t)}\overset{\bigtriangleup}{=}{{{\exp\left( {Q_{i}\left( {t - t_{i}^{e}} \right)} \right)}{x_{i}\left( t_{i}^{e} \right)}} + {\frac{B_{i}}{Q_{i}}\left( {1 - {\exp\left( {Q_{i}\left( {t - t_{i}^{e}} \right)} \right)}} \right)}}},{{{where}B_{i}}\overset{\bigtriangleup}{=}{{\sum_{i\sim j}{\overset{\hat{}}{x}}_{j}} + {k_{i}r}}},{Q_{i} = {k_{i} + d_{i}}},$t_(i) ^(e) is a time when the norm-free information exchange triggeringcondition is violated.
 31. The agent of claim 17, wherein the one ormore neighboring states comprises one or more solution-predictor curvescorresponding to the one or more neighboring vehicles.
 32. The agent ofclaim 17, wherein the last broadcast state is a time-varying real value.